Starting from the Bellman equations we can define two mappings, \(T^{\pi }: \mathbb {R}^n \rightarrow \mathbb {R}^n\)  and \(T: \mathbb {R}^n \rightarrow \mathbb {R}^n\) , where \(T^{\pi }V = g^{\pi } + \gamma P^{\pi }V\)  and \(TV = \min _{\pi \in \Pi }\left\lbrace  g^{\pi } + \gamma P^{\pi } V \right\rbrace \) .
These mappings are known as the Bellman operators and allow one to rewrite (REF ) and (REF ) in a compact form, \(V^{\pi } = T^{\pi }V^{\pi }\)  and \(V^* = TV^*\) , respectively. The Bellman operators are \(\gamma \) -contractive, monotone and shift-invariant and, in our setting, have \(V^{\pi }\)  and \(V^*\)  as their unique fixed-points, respectively. We refer to [1]} for a detailed discussion on the properties of the Bellman operators.
